Wikipedia page for open set gives two definitions for the open set:
First A subset $U$ of the Euclidean $n$-space $\mathbb R^n$ is called open if, given any point $x$ in $U$, there exists a real number $ε > 0$ such that, given any point $y$ in $\mathbb R^n$ whose Euclidean distance from $x$ is smaller than $ε$, $y$ also belongs to $U$.
Second All elements of $T$ where
The trivial subsets are in T.
Whenever sets $A$ and $B$ are in $T$, then so is $A$ intersection $B$.
Whenever two or more sets are in $T$, then so is their union
Consider the topology of $\mathbb R$.
Now, according to the first definition, $[1,5]$ is not open because there exists no ε>0 such that $5+ε$ is in $[1,5]$.
However, we now let $T$ be the family of all subsets of $\mathbb R$.
We can see that $T$ satisfies all $3$ conditions outlined in the second definition. Thus, all elements of $T$ are open, which includes $[1,5]$.
Now, obviously one of these must be wrong. It should be the second one, but I don't see why exactly. Any help is appreciated.
The first definition uses the standard topology on $\mathbb{R}$, while the second is the more general definition of a topology. In your example, you simply created a different topology (the discrete topology) on $\mathbb{R}$.